The MRE inverse problem for the elastic shear modulus

Penny J. Davies, Eric Barnhill, Ingolf Sack

Research output: Contribution to journalArticle

Abstract

Magnetic resonance elastography (MRE) is a powerful technique for noninvasive determination of the biomechanical properties of tissue, with important applications in disease diagnosis. A typical experimental scenario is to induce waves in the tissue by time-harmonic external mechanical oscillation and then measure the tissue's displacement at fixed spatial positions 8 times during a complete time-period, extracting the dominant frequency signal from the discrete Fourier transform in time. Accurate reconstruction of the tissue's elastic moduli from MRE data is a challenging inverse problem, and we derive and analyze two new methods which address different aspects. The first of these concerns the time signal: using only the dominant frequency component loses information for noisy data and typically gives a complex value for the (real) shear modulus, which is then hard to interpret. Our new reconstruction method is based on the Fourier time-interpolant of the displacement: it uses all the measured information and automatically gives a real value of shear modulus up to rounding error. This derivation is for homogeneous materials, and our second new method (stacked frequency wave inversion, SFWI) concerns the inhomogeneous shear modulus in the time-harmonic case. The underlying problem is ill-conditioned because the coefficient of the shear modulus in the governing equations can be zero or small, and the SFWI approach overcomes this by combining approximations at different frequencies into a single overdetermined matrix--vector equation. Careful numerical tests confirm that both these new algorithms perform well.
LanguageEnglish
Pages1367–1388
Number of pages22
JournalSIAM Journal on Applied Mathematics
Volume79
Issue number4
DOIs
Publication statusPublished - 23 Jul 2019

Fingerprint

Resonance Problem
Magnetic Resonance
Magnetic resonance
Inverse problems
Modulus
Inverse Problem
Elastic moduli
Tissue
Inversion
Harmonic
Discrete Fourier transforms
Discrete Fourier transform
Rounding error
Elastic Modulus
Noisy Data
Interpolants
Governing equation
Oscillation
Scenarios
Zero

Keywords

  • magnetic resonance elastography
  • elasticity
  • biomechanics
  • inverse problem

Cite this

Davies, Penny J. ; Barnhill, Eric ; Sack, Ingolf. / The MRE inverse problem for the elastic shear modulus. In: SIAM Journal on Applied Mathematics . 2019 ; Vol. 79, No. 4. pp. 1367–1388.
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The MRE inverse problem for the elastic shear modulus. / Davies, Penny J.; Barnhill, Eric; Sack, Ingolf.

In: SIAM Journal on Applied Mathematics , Vol. 79, No. 4, 23.07.2019, p. 1367–1388.

Research output: Contribution to journalArticle

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